We introduce a pre-symplectic structure on the space of connections in aG-principal bundle over a four-manifold and a Hamiltonian action on it of the group of gauge transformations that are trivial on the boundary. The moment map is given by the square of curvature so that the 0-level set is the space of flat connections. Thus the moduli space of flat connections is endowed with a pre-symplecti...

Let f : X → Spec(Z) be an arithmetic variety of dimension d ≥ 2 and (H, k) an arithmetically ample Hermitian line bundle on X, that is, a Hermitian line bundle with the following properties: (1) H is f -ample. (2) The Chern form c1(H∞, k) gives a Kähler form on X∞. (3) For every irreducible horizontal subvariety Y (i.e. Y is flat over Spec(Z)), the height ĉ1( (H, k)|Y ) dim Y of Y is positive. ...

Let M be an n-dimensional projective algebraic manifold in certain projective space CP . The hyperplane line bundle of CP restricts to an ample line bundle L on M , which is called a polarization of M . A Kähler metric g is called a polarized metric, if the corresponding Kähler form represents the first Chern class c1(L) of L in H (M,Z). Given any polarized Kähler metric g, there is a Hermitian...

Unexpected complex of jumping lines We prove here the following results : Theorem 1 Let E a rank 2 vector bundle over P3, if C is a reduced irreducible curve of P3 such that EH is unstable for all H ∈ C then C is a line. We define now the set W (E) as the set of planes H such that the restricted bundle EH is unstable (that means non semi-stable). Theorem 2 Let E a rank 2 vector bundle over P3, ...

The bigerbes introduced here give a refinement of the notion 2-gerbes, representing degree four integral cohomology classes space. Defined in terms bisimplicial line bundles, have symmetry with respect to which they form 'bundle 2-gerbes' two ways; this structure replaces higher associativity conditions. We provide natural examples, including Brylinski-McLaughlin bigerbe associated principal G-...

In this paper it is proved that the volumes of the moduli spaces of polarized Calabi-Yau manifolds with respect to Weil-Petersson metrics are rational numbers. Mumford introduce the notion of a good metric on vector bundle over a quasi-projective variety in [10]. He proved that the Chern forms of good metrics define classes of cohomology with integer coefficients on the compactified quasi-proje...

We consider a subclass of the Cowen–Douglas class in which problem deciding whether two operators are similar becomes more manageable. A similarity criterion for is known to be dependent on trace curvature corresponding eigenvector bundles. Unless given eignvector bundle line bundle, computation curvature, general, not so simple as one might hope. By using structure theorem operators, we reduce...

Let A be a smooth affine algebraic variety and A one of the Λ-rings given by the three examples in § 2 (b). We want to define the Vicard variety Pic^Ol) with growth condition. To to this, we will give three definitions of what it means for a holomorphic line bundle L —» A to have order A. It is then a basic theorem that the three definitions are equivalent. Definition A. The holomorphic line bu...

(Algebra) It’s a complex projective manifold C of dimension one. As such, it has a field of rational functions K(C) of transcendence degree one over the field C of scalars. The curve can be recovered from the field as the set of discrete valuation rings of K/C, which has the “natural” structure of a projective manifold. There is a canonical line bundle ωC , whose degree captures the genus via d...

Let S be a compact complex surface of general type, Smin its minimal model gotten by contracting the finite number of (−1)-curves. It is a smooth surface with nef canonical line bundle KSmin . It then can be shown that Smin has at most b2 curves on which the canonical line bundleKSmin restricts to a non-ample bundle (i.e. whose intersection number with KSmin is non positive). These are (−2) smo...